john nash: 1928-2015

The mathematician John Nash recently died as the result of an taxi accident in New Jersey. His importance as a scientist is based mainly on a very productive period in his early 20s before he was crippled by the onset of mental illness. His two main scientific contributions are the discovery of the Nash equilibrium for finite games and an extremely important theorem in differential geometry, which essentially states that there exists a vector space where you can embed a smooth surface without self-intersection in a distance preserving fashion.

Here, I want to focus a little on why Nash’s game theory work became so important. He was not the first to study competition within the framework of game theory. Before Nash, a number of economists and mathematicians had worked on game theory, but they got stuck. Some ran into technical problems and could not get beyond the concept of zero sum games. Other had a lack of imagination. The great mathematician Johnny Von Neumann, for example, focused on cooperative games.

Nash’s approach was simple and deep. If you could write down the pay-off matrix of a game, you can come up with the “best response.” If A follows strategy X1, then B’s best answer is Y1. Similarly, if A plays X2, then B will play Y2. It is not hard to see that you can parameterize this argument – you can continuously move from X1 to X2 and thus the other player will move from Y1 to Y2. The profound idea is that having an equilibrium means that the “curve” connecting X1 to X1 intersects the curve connecting Y1 to Y2 and the intersection always exists due to a super-deep theorem called the Kakutani fixed point theorem.

Nash’s approach is ingenious on many levels. It turned an economic modeling problem into a geometry problem. It is very general and works for any game with a finite number of moves. And it spurred the search for equilibrium concepts in even more general settings.